The Nature of the Wave Function
in the Time Independent
Schrödinger Equation

When Erwin Schrödinger formulated the wave mechanics version of quantum physics in 1926 he did not specify
what the wave function ψ represented. He thought its squared magnitude would represent something physical such as charge
density. Max Born suggested that its squared magnitude represented spatial density of finding the particle near a particular location.
Niels Bohr and his group in Copenhagen concurred and the notion that the wave function represents the intrinsic indeterminancy of
the particle of the system came to be known as the Copenhagen Interpretation.

First consider a particle classically traversing periodic trajectory. Let s be path length and the velocity v be given as a
function of s, say v(s). The period of time spent in an interval of length Δs is Δs/|v(s)|, where |v(s)| is the
average speed of the particle in that interval. As Δs→ds the ratio ds/|v(s)| becomes something in the nature
of a density. If the values of 1/|v(s)| are normalized the resulting function can be considered to be a probability
density function P(s). The normalizing constant is just the time period of the trajectory cycle.

P(s) = 1/(Tv(s))
where ∫ds/|v(s)| = ∫dt = T

The kinetic energy of the particle is given by

K = ½mv²
and thus
v = (2K/m)½and henceP(s) = (m/2)&frac12/K&frac12

Any constant multiplier is irrelevant in determining the probability density because it shows up in the normalizing constant as well and
cancels out. The probability density for a classical particle is inversely proportional to the square root of the kinetic energy.

The Quantum Mechanical Wave Function

The time independent Schrödinger equation for a particle in a potential field V(r) is

−h²∇²ψ + V(r)ψ = Eψ

where E is energy and ψ is the wave function, the nature of which is at issue.

This equation can be multiplied on both sides by ψ and rearranged to give

h²ψ∇²ψ = −(E−V(r))ψ²
or, equivalentlyh²ψ∇²ψ = −K(r)ψ²

where K(r) is the kinetic energy of the particle expressed as a function
of the radial distance from the center of the potential field.

ψ∇ψ = 0which is where either
ψ = 0 and ψ² is a minimum of 0
or
∇ψ = 0 and
ψ² is a relative maximum

When the RHS of this equation is substituted for its LHS in the last modified version of the time independent Schrödinger
equation the result is

h²(∇·(ψ∇ψ) −
(1/4)∇(ψ²)·∇(ψ²)/ψ²) = −K(r)ψ²

There is a looped chain of maxima that are separated by minima of zero. This has to be established in
general but for the case of a particle in a potential field the polar coordinates, as is shown in the Appendix, lead to an angular component
of ψ² as shown below.

This is ψ² at a constant value of r and for a principal quantum number of 6.

Below is a depiction of the probability bumps in the plane of the particle's motion for a principal quantum number of 6.

The particle moves quantum mechanically relatively slowly in a probability bump, otherwise known as a state, and then relatively rapidly to the next state (bump).

The analysis now returns to the equation previously developed

h²(∇·(ψ∇ψ) −
(1/4)∇(ψ²)·∇(ψ²)/ψ²) = −K(r)ψ²

If this equation is integrated from one maximum to the next or from one maximum to an adjacent minimum, the first term in
the above equation evaluates to zero because at the end points of the integration either ψ is zero or ∇ψ is the zero vector.
What is left is

where ψ*, ψ# are values of ψ within the interval of integration. The term ∇(ψ²)·∇(ψ²) stands
for the average of ∇(ψ²)·∇(ψ²) in the interval of integration. Multiplying by ψ*² and taking the geometric
mean of ψ*sup2; and ψ#² and denoting it as ψ² gives the equation

(∂²/∂r²) + (1/r)(∂/∂r) + (1/r²)((∂²/∂θ²)

At this point it will be assumed that ψ(r, θ) is equal to R(r)Θ(θ). This is the separation of
variables assumption. This is a mathematical convenience that is fraught with danger of precluding the physically
relevant solutions. In this case it is alright because only circular orbits will be dealt with later.

When R(r)Θ(θ) is substituted into the equation it can be reduced to

−(h²/2m)[R"(r)/R + (1/r)R'(r) + (1/r²)(Θ"(θ)/Θ] + (|E| − Q/r) = 0

This equation may be put into the form

r²R"(r)/R + rR'(r)/R + (2m/(h²)(−|E|r² + rQ) = − (Θ"(θ)/Θ)

The LHS of the above is a function only of r and the RHS a function only of θ. Therefore their common value
must be a constant. Let this constant be denoted as n².

Thus

(Θ"(θ)/Θ) = −n²
or, equivalently
Θ"(θ) + n²Θ(θ)

This equation has solutions of the form

Θ(θ) = A·cos(n·θ + θ0)

where A and θ0 are constants. Through a proper orientation of the polar coordinate system θ0
can made equal to zero. So Θ(θ) = A·cos(n·θ). In order for Θ(θ+2π) to be
equal to Θ(θ) n must be an integer. The probability density is the squared magnitude of the wave function. Therefore the
probability density is proportional to cos²(nθ).